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A359909 Number of integer factorizations of n into factors > 1 with the same mean as median.

%I A359909 #13 Jan 20 2025 09:07:46
%S A359909 0,1,1,2,1,2,1,3,2,2,1,3,1,2,2,4,1,3,1,3,2,2,1,5,2,2,3,3,1,4,1,4,2,2,
%T A359909 2,6,1,2,2,4,1,4,1,3,3,2,1,6,2,3,2,3,1,4,2,4,2,2,1,7,1,2,3,7,2,4,1,3,
%U A359909 2,4,1,7,1,2,3,3,2,4,1,6,4,2,1,6,2,2,2,4,1,6,2,3,2,2,2,6,1,3,3,6,1,4,1,4,5,2,1,6,1,4,2,5,1,4,2,3,3,2,2,11
%N A359909 Number of integer factorizations of n into factors > 1 with the same mean as median.
%C A359909 The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).
%H A359909 Antti Karttunen, <a href="/A359909/b359909.txt">Table of n, a(n) for n = 1..65537</a>
%e A359909 The a(n) factorizations for n = 24, 36, 60, 120, 144, 360:
%e A359909   24      36        60      120       144       360
%e A359909   3*8     4*9       2*30    2*60      2*72      4*90
%e A359909   4*6     6*6       3*20    3*40      3*48      5*72
%e A359909   2*12    2*18      4*15    4*30      4*36      6*60
%e A359909   2*3*4   3*12      5*12    5*24      6*24      8*45
%e A359909           2*2*3*3   6*10    6*20      8*18      9*40
%e A359909                     3*4*5   8*15      9*16      10*36
%e A359909                             10*12     12*12     12*30
%e A359909                             4*5*6     2*2*6*6   15*24
%e A359909                             2*6*10    3*3*4*4   18*20
%e A359909                             2*3*4*5             2*180
%e A359909                                                 3*120
%e A359909                                                 2*10*18
%e A359909                                                 3*4*5*6
%t A359909 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&, Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A359909 Table[Length[Select[facs[n],Mean[#]==Median[#]&]],{n,100}]
%o A359909 (PARI)
%o A359909 median(lista) = if((#lista)%2, lista[(1+#lista)/2], (lista[#lista/2]+lista[1+(#lista/2)])/2);
%o A359909 A359909(n, m=n, facs=List([])) = if(1==n, (#facs>0 && (median(facs)==(vecsum(Vec(facs))/#facs))), my(s=0, newfacs); fordiv(n, d, if((d>1)&&(d<=m), newfacs = List(facs); listput(newfacs,d); s += A359909(n/d, d, newfacs))); (s)); \\ _Antti Karttunen_, Jan 20 2025
%Y A359909 The version for partitions is A240219, complement A359894.
%Y A359909 These multisets are ranked by A359889.
%Y A359909 The version for strict partitions is A359897.
%Y A359909 The odd-length case is A359910.
%Y A359909 The complement is counted by A359911.
%Y A359909 A001055 counts factorizations.
%Y A359909 A058398 counts partitions by mean, see also A008284, A327482.
%Y A359909 A326622 counts factorizations with integer mean, strict A328966.
%Y A359909 A359893 and A359901 count partitions by median, odd-length A359902.
%Y A359909 Cf. A316313, A326567/A326568, A359906, A360005.
%K A359909 nonn
%O A359909 1,4
%A A359909 _Gus Wiseman_, Jan 24 2023
%E A359909 More terms from _Antti Karttunen_, Jan 20 2025